A way to build a conformal boundary of a spacetime based on light - - PowerPoint PPT Presentation

A way to build a conformal boundary of a spacetime based on light rays: the 3–dimensional case.

Alfredo Bautista Santa–Cruz (UC3M-UAM)

co-work with Alberto Ibort (ICMAT-UC3M) and Javier Lafuente (UCM)

ICMAT – Madrid, 9/3/2018 60 years Alberto Ibort Fest Classical and Quantum Physics: Geometry, Dynamics & Control

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Contents

1 Introduction. 2 The space of light rays. 3 The L–boundary for dimension m = 3. 4 L–extensions for dimension m = 3.

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Introduction

1 Introduction. 2 The space of light rays. 3 The L–boundary for dimension m = 3. 4 L–extensions for dimension m = 3.

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Introduction

• R. Low proposed a real geometry based in the space of light rays N ([Low

’88, ’89, ’90, ’93, ’01, ’06]) as a generalization of the twistor geometry ([Penrose ’77, et al ’88]).

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Introduction

• R. Low proposed a real geometry based in the space of light rays N ([Low

’88, ’89, ’90, ’93, ’01, ’06]) as a generalization of the twistor geometry ([Penrose ’77, et al ’88]).

Aim

Construction and characterization of the conformal boundary suggested by

• R. Low, called L–boundary ([Low ’06]), for 3–dimensional spacetimes.
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Introduction

Starting point

(M, Cg) conformal (Lorentz) manifold.

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Introduction

Starting point

(M, Cg) conformal (Lorentz) manifold. M m–dimensional Hausdorff differentiable manifold, (m ≥ 3). g Lorentz metric in M, (− + + + . . . +). (M, g) time–oriented. Cg =

• g = e2σg : σ ∈ F (M)
• conformal (Lorentz) structure in M.
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Introduction

Causality

Given 0 = v ∈ TpM, then v is said to be timelike ⇐ ⇒ g (v, v) < 0 null or lightlike ⇐ ⇒ g (v, v) = 0 spacelike ⇐ ⇒ g (v, v) > 0 causal ⇐ ⇒ timelike or null

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Introduction

Causality

Given 0 = v ∈ TpM, then v is said to be timelike ⇐ ⇒ g (v, v) < 0 null or lightlike ⇐ ⇒ g (v, v) = 0 spacelike ⇐ ⇒ g (v, v) > 0 causal ⇐ ⇒ timelike or null Causal character can be extended to differentiable curves γ : I → M depending on γ′ (s).

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Introduction

Causality

Given 0 = v ∈ TpM, then v is said to be timelike ⇐ ⇒ g (v, v) < 0 null or lightlike ⇐ ⇒ g (v, v) = 0 spacelike ⇐ ⇒ g (v, v) > 0 causal ⇐ ⇒ timelike or null Causal character can be extended to differentiable curves γ : I → M depending on γ′ (s). Causality is conformal.

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Introduction

Figure: Causal character.

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The space of light rays

1 Introduction. 2 The space of light rays. 3 The L–boundary for dimension m = 3. 4 L–extensions for dimension m = 3.

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The space of light rays N

Given g = e2σg ∈ Cg, and a differentiable curve γ : I → M, it is known that γ is null g–geodesic = ⇒ γ is null g–pregeodesic. ([Kulkarni ’88])

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The space of light rays N

Given g = e2σg ∈ Cg, and a differentiable curve γ : I → M, it is known that γ is null g–geodesic = ⇒ γ is null g–pregeodesic. ([Kulkarni ’88])

The space of light rays N

N = {Im (γ) : γ is a null geodesic for some g ∈ Cg}

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The space of light rays N

Given g = e2σg ∈ Cg, and a differentiable curve γ : I → M, it is known that γ is null g–geodesic = ⇒ γ is null g–pregeodesic. ([Kulkarni ’88])

The space of light rays N

N = {Im (γ) : γ is a null geodesic for some g ∈ Cg} This is NOT true for timelike or spacelike geodesics.

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The space of light rays N

Given g = e2σg ∈ Cg, and a differentiable curve γ : I → M, it is known that γ is null g–geodesic = ⇒ γ is null g–pregeodesic. ([Kulkarni ’88])

The space of light rays N

N = {Im (γ) : γ is a null geodesic for some g ∈ Cg} This is NOT true for timelike or spacelike geodesics. A light ray γ ∈ N can be seen as an unparametrized null geodesic. The definition of N is conformal.

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Differentiable structure of N

Strong causality, ([Minguzzi, S´ anchez ’08])

M is strongly causal in x ∈ M if and only if there exists a neighbourhood basis at x of globally hyperbolic, causally convex (and convex normal)

• pen sets.
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Differentiable structure of N

Strong causality, ([Minguzzi, S´ anchez ’08])

M is strongly causal in x ∈ M if and only if there exists a neighbourhood basis at x of globally hyperbolic, causally convex (and convex normal)

• pen sets.

Figure: How can a light ray be defined by a basic neighbourhood?

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Differentiable structure of N

(M, Cg) strongly causal.

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Differentiable structure of N

(M, Cg) strongly causal. TM tangent bundle. TM = TM − {0}.

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Differentiable structure of N

(M, Cg) strongly causal. TM tangent bundle. TM = TM − {0}. N =

• v ∈

TM : g (v, v) = 0

• .
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Differentiable structure of N

(M, Cg) strongly causal. TM tangent bundle. TM = TM − {0}. N =

• v ∈

TM : g (v, v) = 0

• . N = N+ ∪ N−.
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Differentiable structure of N

(M, Cg) strongly causal. TM tangent bundle. TM = TM − {0}. N =

• v ∈

TM : g (v, v) = 0

• . N = N+ ∪ N−.

V ⊂ M basic neighbourhood of x ∈ M contained in a coordinate chart with Cauchy surface C ⊂ V .

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Differentiable structure of N

(M, Cg) strongly causal. TM tangent bundle. TM = TM − {0}. N =

• v ∈

TM : g (v, v) = 0

• . N = N+ ∪ N−.

V ⊂ M basic neighbourhood of x ∈ M contained in a coordinate chart with Cauchy surface C ⊂ V . N+ (C) = {v ∈ N+ : π (v) ∈ C}. (Null vectors at C).

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Differentiable structure of N

(M, Cg) strongly causal. TM tangent bundle. TM = TM − {0}. N =

• v ∈

TM : g (v, v) = 0

• . N = N+ ∪ N−.

V ⊂ M basic neighbourhood of x ∈ M contained in a coordinate chart with Cauchy surface C ⊂ V . N+ (C) = {v ∈ N+ : π (v) ∈ C}. (Null vectors at C). PN (C) = {[v] = span{v} : v ∈ N+ (C)}. (Null directions at C). The topology and the differentiable structure of N is inherited from PN (C) by the diffeomorphism γ : PN (C) → NV given by γ ([v]) = γ[v]

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Differentiable structure of N

(M, Cg) strongly causal. TM tangent bundle. TM = TM − {0}. N =

• v ∈

TM : g (v, v) = 0

• . N = N+ ∪ N−.

V ⊂ M basic neighbourhood of x ∈ M contained in a coordinate chart with Cauchy surface C ⊂ V . N+ (C) = {v ∈ N+ : π (v) ∈ C}. (Null vectors at C). PN (C) = {[v] = span{v} : v ∈ N+ (C)}. (Null directions at C). The topology and the differentiable structure of N is inherited from PN (C) by the diffeomorphism γ : PN (C) → NV given by γ ([v]) = γ[v] Locally, N can be seen as a bundle of spheres: NV ≃ C × Sm−2.

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Topology in N

Definition

M is said to be null pseudo–convex if ∀K compact ∃K ′ compact such that all segments of null geodesic with endpoints at K are contained in K ′.

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Topology in N

Definition

M is said to be null pseudo–convex if ∀K compact ∃K ′ compact such that all segments of null geodesic with endpoints at K are contained in K ′.

Theorem, ([Low ’90])

Let M be strongly causal. M null pseudo–convex ⇐ ⇒ N Hausdorff.

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Topology in N

Definition

M is said to be null pseudo–convex if ∀K compact ∃K ′ compact such that all segments of null geodesic with endpoints at K are contained in K ′.

Theorem, ([Low ’90])

Let M be strongly causal. M null pseudo–convex ⇐ ⇒ N Hausdorff.

Figure: N is not Hausdorff.

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Topology in N

Definition

M is said to be null pseudo–convex if ∀K compact ∃K ′ compact such that all segments of null geodesic with endpoints at K are contained in K ′.

Theorem, ([Low ’90])

Let M be strongly causal. M null pseudo–convex ⇐ ⇒ N Hausdorff.

Figure: M is not null pseudo–convex.

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Topology in N

Definition

M is said to be null pseudo–convex if ∀K compact ∃K ′ compact such that all segments of null geodesic with endpoints at K are contained in K ′.

Theorem, ([Low ’90])

Let M be strongly causal. M null pseudo–convex ⇐ ⇒ N Hausdorff.

Figure: M is not null pseudo–convex.

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Tangent space TγN

How can TγN be described with elements of M?

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Tangent space TγN

How can TγN be described with elements of M?

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Tangent space TγN

How can TγN be described with elements of M?

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Tangent space TγN

How can TγN be described with elements of M?

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Tangent space TγN

How can TγN be described with elements of M?

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Tangent space TγN

How can TγN be described with elements of M?

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Tangent space TγN

How can TγN be described with elements of M?

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Tangent space TγN

How can TγN be described with elements of M?

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Tangent space TγN

If γs are null g–geodesics = ⇒ g (J (t) , γ′ (t)) = constant ∀t.

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Tangent space TγN

If γs are null g–geodesics = ⇒ g (J (t) , γ′ (t)) = constant ∀t. JL (γ) = {J ∈ J (γ) : g (J (t) , γ′ (t)) = const.} ≃ R2m−1 Jacobi fields of null geodesic variation. J0 (γ) = {J ∈ JL (γ) : J (t) = (αt + β) γ′ (t) , α, β ∈ R}

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Tangent space TγN

If γs are null g–geodesics = ⇒ g (J (t) , γ′ (t)) = constant ∀t. JL (γ) = {J ∈ J (γ) : g (J (t) , γ′ (t)) = const.} ≃ R2m−1 Jacobi fields of null geodesic variation. J0 (γ) = {J ∈ JL (γ) : J (t) = (αt + β) γ′ (t) , α, β ∈ R}

Proposition

TγN is isomorphic to L (γ) = JL (γ) /J0 (γ) ≃ R2m−3. This means: ξ ∈ TγN ⇐ ⇒ J (modγ′) ∈ JL (γ) /J0 (γ) The metric g ∈ C and the parametrization of γ ∈ N are auxiliary elements.

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Contact structure H

Contact structure H in P where dim P = 2n + 1

H is a differentiable distribution of hyperplanes in TP defined by a 1–form α verifying α ∧ (dα)n = 0 (maximally non–integrable). If α is global then H is said to be co–oriented.

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Contact structure H

Contact structure H in P where dim P = 2n + 1

H is a differentiable distribution of hyperplanes in TP defined by a 1–form α verifying α ∧ (dα)n = 0 (maximally non–integrable). If α is global then H is said to be co–oriented.

Contact structure H in N

Hγ = {J ∈ TγN : α (J) ≡ g (J, γ′) = 0}. Moreover, H is co–oriented. H is conformal

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The space of skies Σ

The sky of x ∈ M

The sky of x is defined by X = S (x) = {γ ∈ N : x ∈ γ} ≃ Sm−2

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The space of skies Σ

The sky of x ∈ M

The sky of x is defined by X = S (x) = {γ ∈ N : x ∈ γ} ≃ Sm−2

The space (set) of skies

Σ = {S (x) : x ∈ M}

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The space of skies Σ

The sky of x ∈ M

The sky of x is defined by X = S (x) = {γ ∈ N : x ∈ γ} ≃ Sm−2

The space (set) of skies

Σ = {S (x) : x ∈ M}

The sky map

S : M → Σ defined by x → X = S (x)

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The space of skies Σ

The sky of x ∈ M

The sky of x is defined by X = S (x) = {γ ∈ N : x ∈ γ} ≃ Sm−2

The space (set) of skies

Σ = {S (x) : x ∈ M}

The sky map

S : M → Σ defined by x → X = S (x) S is surjective by definition. Injectivity of S is required

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The space of skies Σ

The sky of x ∈ M

The sky of x is defined by X = S (x) = {γ ∈ N : x ∈ γ} ≃ Sm−2

The space (set) of skies

Σ = {S (x) : x ∈ M}

The sky map

S : M → Σ defined by x → X = S (x) S is surjective by definition. Injectivity of S is required ⇐ ⇒ M is sky–separating.

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The space of skies Σ

Figure: Skies at M.

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The space of skies Σ

Figure: Skies at N.

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The space of skies Σ

For any x ∈ M and γ ∈ S (x), if x = γ (t0) then

Tangent space to a sky

TγS (x) = {J ∈ TγN : J (t0) = 0 (mod γ′)} ⊂ Hγ

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The space of skies Σ

For any x ∈ M and γ ∈ S (x), if x = γ (t0) then

Tangent space to a sky

TγS (x) = {J ∈ TγN : J (t0) = 0 (mod γ′)} ⊂ Hγ

Light non-conjugation

TγX ∩ TγY = {0γ} = ⇒ X = Y ∈ Σ

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The space of skies Σ

For any x ∈ M and γ ∈ S (x), if x = γ (t0) then

Tangent space to a sky

TγS (x) = {J ∈ TγN : J (t0) = 0 (mod γ′)} ⊂ Hγ

Light non-conjugation

TγX ∩ TγY = {0γ} = ⇒ X = Y ∈ Σ

H is conformal

Hγ = TγX ⊕ TγY for any X, Y light non–conjugate.

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Topology in Σ

X0 ∈ Σ such that X0 ⊂ U ⊂ N where U is open.

Reconstructive or Low’s topology in Σ

It is generated by Σ (U) = {X ∈ Σ : X ⊂ U}

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Topology in Σ

X0 ∈ Σ such that X0 ⊂ U ⊂ N where U is open.

Reconstructive or Low’s topology in Σ

It is generated by Σ (U) = {X ∈ Σ : X ⊂ U}

Theorem ([Kinlaw ’11], [–, Ibort, Lafuente 14, 15])

The sky map S : M → Σ is an homeomorphism.

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Differentiable structure in Σ

Theorem ([–, Ibort, Lafuente 14, 15])

Let V ⊂ M be a relatively compact basic open set and VΣ = S (V ) ⊂ Σ. Then: VΣ ⊂ Σ is light non–conjugate.

• V =
• X∈VΣ
• TX ⊂ TN is a regular submanifold of

TN. The distribution D in TN such that its leaves are TX is regular. The map VΣ → V /D such that X → TX is a diffeomorphism.

Theorem ([–, Ibort, Lafuente 14, 15])

The previous one is the unique differentiable structure of Σ compatible with the reconstructive topology and such that S : M → Σ is a diffeomorphism.

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Reconstruction of (M, C)

Theorem of reconstruction ([–, Ibort, Lafuente, ’14])

Let (M, C),

• M, C
• be two strongly causal spacetimes and (N, Σ),
• N, Σ
• their corresponding pairs of spaces of light rays and skies. Let φ : N → N

be a diffeomorphism such that φ (Σ) ⊂ Σ (i.e. sky preserving). Then the map ϕ = S

−1 ◦ φ ◦ S : M → M

is a conformal diffeomorphism onto its image, where S : M → Σ is the sky map of M.

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The L–boundary for dimension m = 3

1 Introduction. 2 The space of light rays. 3 The L–boundary for dimension m = 3. 4 L–extensions for dimension m = 3.

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Low’s boundary

Idea

γ : (a, b) → M maximal parametrized light ray.

• γ : (a, b) → Grm−2 (Hγ) defined by

γ (s) = TγS (γ (s)) where Grm−2 (Hγ) is the grassmannian manifold of (m − 2)–dimensional subspaces of Hγ ⊂ TγN. ⊖γ = lims→a+ γ (s) ∈ Grm−2 (Hγ) ⊕γ = lims→b− γ (s) ∈ Grm−2 (Hγ) (distributions in Grm−2 (H)). New future (past) causal boundary: integral manifolds of ⊕ (⊖).

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Low’s boundary

Idea

γ : (a, b) → M maximal parametrized light ray.

• γ : (a, b) → Grm−2 (Hγ) defined by

γ (s) = TγS (γ (s)) where Grm−2 (Hγ) is the grassmannian manifold of (m − 2)–dimensional subspaces of Hγ ⊂ TγN. ⊖γ = lims→a+ γ (s) ∈ Grm−2 (Hγ) ⊕γ = lims→b− γ (s) ∈ Grm−2 (Hγ) (distributions in Grm−2 (H)). New future (past) causal boundary: integral manifolds of ⊕ (⊖).

Problem

Do the limits ⊖γ and ⊕γ exist?

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Low’s boundary in dim M = 3

dim M = 3 = ⇒ dim N = 3. Gr1 (Hγ) = P (Hγ).

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Low’s boundary in dim M = 3

dim M = 3 = ⇒ dim N = 3. Gr1 (Hγ) = P (Hγ).

Figure: γ ⊂ P (Hγ).

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Low’s boundary in dim M = 3

dim M = 3 = ⇒ dim N = 3. Gr1 (Hγ) = P (Hγ).

Figure: γ ⊂ P (Hγ).

If M is light non–conjugate = ⇒ the limits ⊖γ, ⊕γ exist.

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Hypotheses

Hypotheses

• a. dim M = 3.
• b. (M, C) is strongly causal, null–pseudo convex, light non–conjugate

and sky–separating.

• c. The distributions ⊕, ⊖ : N → P (H) defined by

⊕γ = lims→b− TγS (γ (s)) and ⊖γ = lims→a+ TγS (γ (s)) are differentiable and regular and such that ⊕γ = ⊖γ for any maximally and future–directed parametrized light ray γ : (a, b) → M.

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The space N ⊂ P (H)

Lemma

Let πPN

M : PN → M be the canonical projection. Then the map

σ : PN → P (H) [u] → Tγ[u]S

• πPN

M ([u])

• (1)

is a diffeomorphism onto its image N = σ (PN).

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The space N ⊂ P (H)

Lemma

Let πPN

M : PN → M be the canonical projection. Then the map

σ : PN → P (H) [u] → Tγ[u]S

• πPN

M ([u])

• (1)

is a diffeomorphism onto its image N = σ (PN).

Corollary

• N is an open submanifold of P (H).
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The projective parameter

Every fibre P (Hγ) is a projective line, so it is diffeomorphic to the circle S1.

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The projective parameter

Every fibre P (Hγ) is a projective line, so it is diffeomorphic to the circle S1. We can construct a “bunch” of projective parametrizations ε : NU × R → P (HU) − ∞ (γ, t) →

• γ (t) = TγS (γ (t))

such that ε is a diffeomorphism and moreover t ∈ (−1, 1) ⇐ ⇒ γ (t) ∈ N.

• γ (1) = ⊕γ,

γ (−1) = ⊖γ and π ◦ σ−1 ( γ (0)) ∈ C.

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Distribution D∼

σ : PN → N diffeomorphism. P regular distribution in PN whose leaves are fibres PNx of PN → M. Propagating P by σ we obtain a regular distribution D∼ in N whose leaves are σ (PNx).

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Distribution D∼

σ : PN → N diffeomorphism. P regular distribution in PN whose leaves are fibres PNx of PN → M. Propagating P by σ we obtain a regular distribution D∼ in N whose leaves are σ (PNx). The map PN/P → N/D∼ induced by σ is a diffeomorphism. Since PN/P ≃ M then

• N/D∼ ≃ M
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Distribution ∂+D∼

N ≃ N × {1}

ε

→ ∂+ N is a diffeomorphism. ⊕ : N → P (H) is a regular distribution in N. Propagating the leaves of ⊕ by ε we obtain a regular distribution ∂+D∼ in ∂+ N whose leaves are ε (X +) for X + leaf of ⊕.

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L–boundary

Theorem, ([–, Ibort, Lafuente, ’18])

D∼ = D∼ ∪ ∂+D∼ is a regular smooth distribution in N.

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L–boundary

Theorem, ([–, Ibort, Lafuente, ’18])

D∼ = D∼ ∪ ∂+D∼ is a regular smooth distribution in N. Then

• N/D∼ is a differentiable manifold and
• N/D∼ =

N/D∼ ∪ ∂+ N/∂+D∼ ≃ M ∪ ∂+ N/∂+D∼ = M ∂M = ∂+ N/∂+D∼ is the L–boundary.

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L–extensions for dimension m = 3.

1 Introduction. 2 The space of light rays. 3 The L–boundary for dimension m = 3. 4 L–extensions for dimension m = 3.

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L–extensions

Let γ : (a, b) → M be an inextensible parametrization of a light ray γ ∈ N such that γ ⊂ M is future–directed. This parametrization is said to be

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L–extensions

Let γ : (a, b) → M be an inextensible parametrization of a light ray γ ∈ N such that γ ⊂ M is future–directed. This parametrization is said to be

1 continuous if γ : (a, b) → M is a continuous map,

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L–extensions

Let γ : (a, b) → M be an inextensible parametrization of a light ray γ ∈ N such that γ ⊂ M is future–directed. This parametrization is said to be

1 continuous if γ : (a, b) → M is a continuous map, 2 regular if γ : (a, b) → M is a differentiable map and γ′ (s) ∈ N is a

future–directed lightlike vector for all s ∈ (a, b),

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L–extensions

Let γ : (a, b) → M be an inextensible parametrization of a light ray γ ∈ N such that γ ⊂ M is future–directed. This parametrization is said to be

1 continuous if γ : (a, b) → M is a continuous map, 2 regular if γ : (a, b) → M is a differentiable map and γ′ (s) ∈ N is a

future–directed lightlike vector for all s ∈ (a, b),

3 projective if γ : (a, b) → M is a regular parametrization and

• γ (s) ∈ P (Hγ) defines a projectivity in the fibre P (Hγ), and
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L–extensions

Let γ : (a, b) → M be an inextensible parametrization of a light ray γ ∈ N such that γ ⊂ M is future–directed. This parametrization is said to be

1 continuous if γ : (a, b) → M is a continuous map, 2 regular if γ : (a, b) → M is a differentiable map and γ′ (s) ∈ N is a

future–directed lightlike vector for all s ∈ (a, b),

3 projective if γ : (a, b) → M is a regular parametrization and

• γ (s) ∈ P (Hγ) defines a projectivity in the fibre P (Hγ), and

4 admissible if there exists a diffeomorphism h : (c, d] → (a, b] such

that h′ (t) > 0 for all t ∈ (c, d] and γ ◦ h : (c, d) → M is a projective parametrization.

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L–extensions

Definition

A future L–extension of (M, C) is a Hausdorff smooth manifold M = M ∪ ∂+M where ∂+M = M − M is a closed hypersurface of M called the future L–boundary such that:

1 lims→b− γ (s) = ∞+

γ ∈ ∂+M for any continuous parametrization of

γ ∈ N.

2 The map ∞+ : N → ∂+M defined by ∞+ (γ) = ∞+

γ is a surjective

submersion.

3 For every γ0 ∈ N there exists a neighbourhood U ⊂ N and a

differentiable map ΨU : U × (a, b] → M, where γ (s) = ΨU (γ, s) is an admissible parametrization of γ ∈ U for s ∈ (a, b) and such that

∂ΨU ∂s (γ, b) /

∈ T∞+(γ)∂+M. Analogously, we can define a past L–extension M = M ∪ ∂−M.

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L–extensions

Since ∞+ : N → ∂+M is a surjective submersion then every S (p) =

• ∞+−1 (p) = {γ ∈ N : p = ∞+ (γ)} ⊂ N

defines a leaf of a regular distribution ⊞ : N → P (TN) given by ⊞ (γ) = TγS (∞+ (γ)), and the map S : ∂+M → N/⊞ p → S (p) is a diffeomorphism.

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L–extensions

Since ∞+ : N → ∂+M is a surjective submersion then every S (p) =

• ∞+−1 (p) = {γ ∈ N : p = ∞+ (γ)} ⊂ N

defines a leaf of a regular distribution ⊞ : N → P (TN) given by ⊞ (γ) = TγS (∞+ (γ)), and the map S : ∂+M → N/⊞ p → S (p) is a diffeomorphism.

Theorem, ([–, Ibort, Lafuente, ’18])

The extension constructed by the L–boundary is a L–extension.

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L–extensions

Theorem, ([–, Ibort, Lafuente, ’18])

Let M1 = M ∪ ∂+M1 and M2 = M ∪ ∂+M2 be future L–extension of (M, C), then the identity map id : M → M can be extended as a diffeomorphism id : M1 → M2.

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L–extensions

Theorem, ([–, Ibort, Lafuente, ’18])

Let M1 = M ∪ ∂+M1 and M2 = M ∪ ∂+M2 be future L–extension of (M, C), then the identity map id : M → M can be extended as a diffeomorphism id : M1 → M2. Conversely,

Theorem, ([–, Ibort, Lafuente, ’18])

Under the hypotheses:

• a. dim M = 3.
• b. (M, C) is strongly causal, null–pseudo convex, light non–conjugate

and sky–separating.

• c. there is a future L–extension of (M, C)

then ⊞ = ⊕, and therefore ⊕ is differentiable and regular.

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Present and future

Present

Construction and characterization of L–boundary for 3–dimensional spacetimes. All results try to indicate, when possible, a way to afford the construction for the general higher dimensional case.

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Present and future

Present

Construction and characterization of L–boundary for 3–dimensional spacetimes. All results try to indicate, when possible, a way to afford the construction for the general higher dimensional case.

What’s next?

Follow the suggested way (or find another one) to obtain analogue results for m–dimensional spacetimes with m ≥ 3.

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Main bibliography

– , A. Ibort, J. Lafuente, On the space of light rays of a spacetime and a reconstruction theorem by Low. Class. Quant. Grav. 31 (2014) 075020. – , A. Ibort, J. Lafuente, Causality and skies: is non-refocussing necessary?. Class. Quant. Grav. 32 (2015) 105002. – , A. Ibort, J. Lafuente, R. Low. A conformal boundary for space-times based on light-like geodesics: The 3–dimensional case. J.

• Math. Phys. 58 (2017) 022503.

Low, R.J. Stable singularities of wave-fronts in general relativity. J.

• Math. Phys. 39 (6), 3332–3335 (1998).

Low, R.J. The space of null geodesics (and a new causal boundary). Lecture Notes in Physics 692 35–50 (2006).

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Thanks to all for your attention... ... and congratulations to Alberto for his birthday. Happy ✟✟

✟ ❍❍ ❍

sixty sixteen!!!

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